Fits MC fractions to data histogram (a la HMCMLL, see R. Barlow and C. Beeston,
Comp. Phys. Comm. 77 (1993) 219-228, and http://www.hep.man.ac.uk/~roger/hfrac.f).
The virtue of this fit is that it takes into account both data and Monte Carlo
statistical uncertainties. The way in which this is done is through a standard
likelihood fit using Poisson statistics; however, the template (MC) predictions
are also varied within statistics, leading to additional contributions to the
overall likelihood. This leads to many more fit parameters (one per bin per
template), but the minimisation with respect to these additional parameters is
done analytically rather than introducing them as formal fit parameters. Some
special care needs to be taken in the case of bins with zero content. For more
details please see the original publication cited above.
An example application of this fit is given below. For a TH1* histogram
("data") fitted as the sum of three Monte Carlo sources ("mc"):
{
TH1F *data; //data histogram
TH1F *mc0; // first MC histogram
TH1F *mc1; // second MC histogram
TH1F *mc2; // third MC histogram
.... // retrieve histograms
TObjArray *mc = new TObjArray(3); // MC histograms are put in this array
mc->Add(mc0);
mc->Add(mc1);
mc->Add(mc2);
TFractionFitter* fit = new TFractionFitter(data, mc); // initialise
fit->Constrain(1,0.0,1.0); // constrain fraction 1 to be between 0 and 1
fit->SetRangeX(1,15); // use only the first 15 bins in the fit
Int_t status = fit->Fit(); // perform the fit
cout << "fit status: " << status << endl;
if (status == 0) { // check on fit status
TH1F* prediction = (TH1F*) fit->GetPlot();
data->Draw("Ep");
result->Draw("same");
}
}
Instantiation
=============
A fit object is instantiated through
TFractionFitter* fit = new TFractionFitter(data, mc);
A number of basic checks (intended to ensure that the template histograms
represent the same "kind" of distribution as the data one) are carried out.
The TVirtualFitter object is then addressed and all fit parameters (the
template fractions) declared (initially unbounded).
Applying constraints
====================
Fit parameters can be constrained through
fit->Constrain(parameter #, lower bound, upper bound);
Setting lower bound = upper bound = 0 removes the constraint (a la Minuit);
however, a function
fit->Unconstrain(parameter #)
is also provided to simplify this.
Setting parameter values
========================
The function
TVirtualFitter* vFit = fit->GetFitter();
is provided for direct access to the TVirtualFitter object. This allows to
set and fix parameter values, and set step sizes directly.
Restricting the fit range
=========================
The fit range can be restricted through
fit->SetRangeX(first bin #, last bin #);
and freed using
fit->ReleaseRangeX();
For 2D histograms the Y range can be similarly restricted using
fit->SetRangeY(first bin #, last bin #);
fit->ReleaseRangeY();
and for 3D histograms also
fit->SetRangeZ(first bin #, last bin #);
fit->ReleaseRangeZ();
Weights histograms
==================
Weights histograms (for a motivation see the above publication) can be specified
for the individual MC sources through
fit->SetWeight(parameter #, pointer to weights histogram);
and unset by specifying a null pointer.
Obtaining fit results
=====================
The fit is carried out through
Int_t status = fit->Fit();
where status is the code returned from the "MINIMIZE" command. For fits
that converged, parameter values and errors can be obtained through
fit->GetResult(parameter #, value, error);
and the histogram corresponding to the total Monte Carlo prediction (which
is not the same as a simple weighted sum of the input Monte Carlo distributions)
can be obtained by
TH1* result = fit->GetPlot();
Using different histograms
==========================
It is possible to change the histogram being fitted through
fit->SetData(TH1* data);
and to change the template histogram for a given parameter number through
fit->SetMC(parameter #, TH1* MC);
This can speed up code in case of multiple data or template histograms;
however, it should be done with care as any settings are taken over from
the previous fit. In addition, neither the dimensionality nor the numbers of
bins of the histograms should change (in that case it is better to instantiate
a new TFractionFitter object).
Errors
======
Any serious inconsistency results in an error.

TFractionFitter constructor. Does a complete initialisation (including
consistency checks, default fit range as the whole histogram but without
under- and overflows, and declaration of the fit parameters). Note that
the histograms are not copied, only references are used.
Arguments:
data: histogram to be fitted
MCs: array of TH1* corresponding template distributions

Change the histogram to be fitted to. Notes:
- Parameter constraints and settings are retained from a possible previous fit.
- Modifying the dimension or number of bins results in an error (in this case
rather instantiate a new TFractionFitter object)

Change the histogram for template number <parm>. Notes:
- Parameter constraints and settings are retained from a possible previous fit.
- Modifying the dimension or number of bins results in an error (in this case
rather instantiate a new TFractionFitter object)

Set bin by bin weights for template number <parm> (the parameter numbering
follows that of the input template vector).
Weights can be "unset" by passing a null pointer.
Consistency of the weights histogram with the data histogram is checked at
this point, and an error in case of problems.

Set the X range of the histogram to be used in the fit.
Use ReleaseRangeX() to go back to fitting the full histogram.
The consistency check ensures that no empty fit range occurs (and also
recomputes the bin content integrals).
Arguments:
low: lower X bin number
high: upper X bin number

Set the Y range of the histogram to be used in the fit (2D or 3D histograms only).
Use ReleaseRangeY() to go back to fitting the full histogram.
The consistency check ensures that no empty fit range occurs (and also
recomputes the bin content integrals).
Arguments:
low: lower Y bin number
high: upper Y bin number

Set the Z range of the histogram to be used in the fit (3D histograms only).
Use ReleaseRangeY() to go back to fitting the full histogram.
The consistency check ensures that no empty fit range occurs (and also
recomputes the bin content integrals).
Arguments:
low: lower Y bin number
high: upper Y bin number

Function used internally to check the consistency between the
various histograms. Checks are performed on nonexistent or empty
histograms, the precise histogram class, and the number of bins.
In addition, integrals over the "allowed" bin ranges are computed.
Any inconsistency results in a error.

Return the "template prediction" corresponding to the fit result (this is not
the same as the weighted sum of template distributions, as template statistical
uncertainties are taken into account).
Note that the name of this histogram will simply be the same as that of the
"data" histogram, prefixed with the string "Fraction fit to hist: ".

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